Persymmetric Adaptive Union Subspace Detection

This paper addresses the detection of a signal belonging to several possible subspace models, namely, a union of subspaces (UoS), where the active subspace that generated the observed signal is unknown. By incorporating the persymmetric structure of received data, we propose three UoS detectors based on GLRT, Rao, and Wald criteria to alleviate the requirement of training data. In addition, the detection statistic and classification bound for the proposed detectors are derived. Monte-Carlo simulations demonstrate the detection and classification performance of the proposed detectors over the conventional detector in training-limited scenarios.

Structure of the set of Borel exceptional vectors for entire curves. II

We have obtained a description of structure of the sets of Picard and Borel exceptional vectors for transcendental entire curve in some sense. We consider only $p$-dimensional entire curves with linearly independent components without common zeros.In particular, the set of Borel exceptional vectors together with the zero vector is a finite union of subspaces in $\mathbb{C}^{p}$ of dimension at most $p-1$. Moreover, the sum of their dimensions does not exceed $p$ if anypairwise intersection of the subspaces contains only the zero vector. A similar result is also valid for the set of Picardexceptional vectors.Another result shows that the structure of the set of Borel exceptional vectors for an entire curve of integer orderdiffers somewhat from the structure of such a set for an entire curve of non-integer order.For a transcendental entire curve $\vec{G}:\mathbb{C}\to \mathbb{C}^{p}$ with linearly independent components and without common zeros having non-integer or zero order the set of Borel exceptional vectors together with the zero vector is a subspace in $\mathbb{C}^{p}$ of dimension at most $p-1$. However, the set of Picard exceptional vectors does not possess this property. We propose two examples of entire curves.The first example shows the set of Borel exceptional vectors together with the zero vector for $p$-dimensional entire curve of integer order isunion of subspaces of dimension at most $p-1$ such that the total sum of these dimensions does not exceed $p$ and intersection of any pair of these subspaces contains only zero vector. The set of Picard exceptional vectors for the curve has the same property.In the second example, we construct a $q$-dimensional entire curve of non-integer order for which the set of Borel exceptional vectors together with the zero vector is a subspace in $\mathbb{C}^{q}$ of dimension at most $q-1$ and the set of Picard exceptional vectors together with the zero vectordo not have the property. This set is a union of some subspaces.

Piecewise Sparse Approximation via Threshold P_OMP with Application to Surface Reconstruction

Sparse signal representations have gained extensive study in recent years. In applications, there are large amounts of signals that are structured. Motivated by signal decomposition and scattered data reconstruction applications, we consider a particular type of structured signals which can be represented by a union of several sparse vectors. We define this type of signal as piecewise sparse signals. To find a piecewise sparse representation of a signal, we propose a thresholding version of piecewise orthogonal matching pursuit(TP OMP), which aims to overcome the disadvantages of P_OMP. We also establish the connection of piecewise sparsity and sampling over a union of subspaces. We evaluate the performance of the proposed greedy programs through simulations on surface reconstruction.